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Multiplying Complex Numbers: (x + 1 + 3i)(x + 1 - 3i)
This expression involves multiplying two complex numbers, which can be approached using the FOIL method (First, Outer, Inner, Last) or by recognizing the pattern of a difference of squares. Let's explore both approaches.
Using FOIL Method
- First: (x)(x) = x²
- Outer: (x)(-3i) = -3ix
- Inner: (1)(x) = x
- Last: (1)(-3i) = -3i
- Middle terms: We also have (3i)(x) = 3ix and (3i)(-3i) = -9i²
Combining all the terms, we get:
x² - 3ix + x - 3i + 3ix - 9i²
Since i² = -1, we can simplify further:
x² + x - 3i + 3ix + 9
Finally, combining real and imaginary terms:
x² + x + 9
Using Difference of Squares
We can recognize the given expression as a difference of squares:
(x + 1 + 3i)(x + 1 - 3i) = [(x + 1) + 3i][(x + 1) - 3i]
The difference of squares pattern states: (a + b)(a - b) = a² - b²
Applying this pattern:
[(x + 1) + 3i][(x + 1) - 3i] = (x + 1)² - (3i)²
Expanding and simplifying:
x² + 2x + 1 - 9i² = x² + 2x + 1 + 9
Therefore, (x + 1 + 3i)(x + 1 - 3i) simplifies to x² + 2x + 10.
Key Points
- This problem demonstrates how to multiply complex numbers using the FOIL method and the difference of squares pattern.
- The result of multiplying these complex conjugates is always a real number.
- Understanding complex number operations is essential in various fields, including mathematics, physics, and engineering.